We study the notions of acs, luacs and uacs Banach spaces which were introduced in [26] and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested in (mainly infinite) absolute sums of such spaces. We also introduce some new classes of spaces that lie inbetween those of acs and uacs spaces and study their behaviour under the formation of absolute sums as well.x n + x → 2 ⇒ x n → x weakly,

We prove some results concerning the WORTH property and the García-Falset coefficient of absolute sums of infinitely many Banach spaces. The Opial property/uniform Opial property of infinite ℓ p -sums is also studied and some properties analogous to the Opial property/uniform Opial property for Lebesgue-Bochner spaces L p (µ, X) are discussed.

X. Huang et al. recently introduced the notion of generalised lush (GL) spaces in [13], which, at least for separable spaces, is a generalisation of the concept of lushness introduced in [3]. The main result of [13] is that every GL-space has the so called Mazur-Ulam property (MUP). In this note, we will prove some properties of GL-spaces (further than those already established in [13]), for example, every M -ideal in a GL-space is again a GL-space, ultraproducts of GL-spaces are again GL-spaces, and if the bidual X * * of a Banach space X is GL, then X itself still has the MUP.

Abstract. We study the question whether properties like local/weak almost squareness and local octahedrality pass down from an absolute sum X ⊕F Y to the summands X and Y .

We consider a certain type of geometric properties of Banach spaces, which includes, for instance, octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem, which, roughly speaking, states the following: if the property in question is stable under certain finite absolute sums (for example, finite p -sums), then it is also stable under the formation of corresponding Köthe-Bochner spaces (for example, L p -Bochner spaces). From this general theorem, we obtain as corollaries a number of new results as well as some alternative proofs of already known results concerning octahedral and almost square spaces and their relatives, diameter two properties, lush spaces and other classes.